Roland GeraertsSeminar Crowd Simulation 2011
Path Planning with Explicit Corridor Maps Related work Constructing Explicit Corridor Maps Corridor Map Method Exploiting Explicit Corridors
Related work: A*
Method• Construction phase: create a grid, mark free/blocked cells• Query phase: use A* to find the shortest path (in the grid)
Advantage• Simple
Disadvantages• Too slow in large scenes• Ugly paths
– Little clearance to obstacles– Unnatural motions (sharp turns)
• Fixed paths– Predictable motions
Related work: Potential Fields
Method• Goal generates attractive force• Obstacles generate repulsive force• Follow the direction of steepest
descent of the potential toward the goal
Advantages• Flexibility to avoid local hazards• Smooth paths
Disadvantages• Expensive for multiple goals• Local minima
Related work: Probabilistic Roadmap Method
Method• Construction phase: build the roadmap• Query phase: query the roadmap
Advantages• Reasonably fast• High-dimensional problems
Disadvantages• Ugly paths• Fixed paths
– Predictable motions– Lacks flexibility when
environment changes
Related work: Probabilistic Roadmap Method
Method• Construction phase: build the roadmap• Query phase: query the roadmap
Advantages• Reasonably fast• High-dimensional problems
Disadvantages• Ugly paths• Fixed paths
– Predictable motions– Lacks flexibility when
environment changes
State-of-the-art: Navigation meshes
Method• Create a representation of the
"walkable areas" of an environment• Extract the path
Advantages• General approach• Construction is fast due to use of GPU
– Examples and source code can be found on http://code.google.com/p/recastnavigation
Disadvantages• Often needs a lot of manual editing• Current techniques are imprecise• Bad support for non-planar surfaces
Obstacles Walkable voxels
Voxel regions Polygonal regions
Convex regions A path
State-of-the-art: Navigation meshes
Some open problems• Automatic annotation of the map
– Areas: walk, climb, jump, crouch, “avoid” …– Special places: hiding and sniper spots, …
• Handle large (dynamic) changes– Efficiently updating the data structure and paths
• Improve the efficiency of mesh generation (large scenes)• Wrong coverage/connectivity due to confusing elements
– Steep stairs, ramps, hills, curved surfaces, gaps
• The mesh is only a data structure storing the walkable areas– How to create visually convincing paths?
Towards a new methodology: Requirements
Fast and flexible path planner• Real-time planning for thousands of characters• Dealing with local hazards• Global path
Natural paths• Smooth• Short• Keeps some distance to
obstacles• Avoids other characters• …
Titan Quest: Immortal throne
Capturing the free space
Requirements of the data structure representing the free (walkable) space• Existence of a path• Contains all cycles
– Short global paths, alternative paths
• Provides high-clearance paths (corridors)– Provides maximum local flexibility
• Small size• Fast extraction of paths
A good candidate• Generalized Voronoi Diagram + annotation
Voronoi Diagram
Some inspiration from natural objects…drying mud maple leaf
wasps nestgiraffe bacteria colonies
Voronoi Diagram
Definitions• Voronoi region: set of all points closest to a given point• Voronoi diagram: union of all Voronoi regions
Voronoi sites: (red) points
Voronoi Diagram
Approximation of the Voronoi Diagram• Compute a distance mesh for each point• Render each mesh in a different color by using the GPU
– Using the Z-buffer, only pixels with the lowest distance values attribute to a pixel in the Frame buffer
• A parallel projection of the meshes gives the diagram
Perspective view (Z-buffer) Top view (Frame buffer)
Generalized Voronoi Diagram
Generalized Voronoi Diagram supports any type of obstacles• Point, disk, line, polygon, …• Convert concave polygons into convex ones, otherwise edges do not
run into all corners
Generalized Voronoi Diagram
Generalized Voronoi Diagram supports any type of obstacles• Point, disk, line, polygon, …• Convert concave polygons into convex ones, otherwise edges do not
run into all corners
Distance meshes• Point: cone• Disk: lifted cone• Line: tent + 2 cones• Polygon: n (point + line meshes)
Literature• [Hoff et al., 1999]• [Geraerts and Overmars, 2010]
From GDV to Medial axis
Generalized Voronoi Diagram (GVD)• Render distance meshes for each obstacle• Boundaries: bisectors between any two closest obstacles
Medial axis• Yields bisectors between any two distinct closest obstacles• Extraction of the medial axis
– Edge: trace pixels between Voronoi regions; continue tracingwhen closest points on the obstacles are equal
– Vertex: end point of an edge
GVD Medial axis
Medial axis
The good• Existence of a path: full coverage/connectivity• Contains all cycles: yes• Provides high-clearance paths: yes• Small size: yes (linear)• Fast extraction of paths: yes
The bad• Unclear how to extract short(est) paths• Moving along 1D-curves limits flexibility
The ugly• Deal with robustness
Explicit Corridor Map
Basis: Medial Axis Plus: annotated event points on the edges
• Points where the type of bisector on the edge changes– Straight lines versus parabola’s (bisector of point and line)
• Changes occur at crossing between site normal and edge• Annotation: its two closest points on the sites
Equals: planar subdivision (or navigation mesh)Memory footprint The storage is linear in the number of obstacle vertices.
Memory footprint The storage is linear in the number of obstacle vertices.
Note There is no need for storing pixels.
Note There is no need for storing pixels.
Explicit Corridor Map: closest points
Computation of the closest points• Look up incident colors at the event point’s position• Each color was linked to an unique obstacle• Compute the (left and right) closest points to each obstacle
using simple linear algebra
Explicit Corridor Map: experiments
Performance• Setup
– NVIDIA GeForce 8800 GTX graphics card– Intel Core2 Quad CPU 2.4 GHz, 1 CPU used
• Experiments– McKenna: 200x200 meter, 1600x1600 pixels, 23 convex polygons
Explicit Corridor Map: experiments
Performance• Setup
– NVIDIA GeForce 8800 GTX graphics card– Intel Core2 Quad CPU 2.4 GHz, 1 CPU used
• Experiments– McKenna: 200x200 meter, 1600x1600 pixels, 23 convex polygons
time: 0.03s
Explicit Corridor Map: experiments
Performance• Setup
– NVIDIA GeForce 8800 GTX graphics card– Intel Core2 Quad CPU 2.4 GHz, 1 CPU used
• Experiments– City: 500x500 meter, 4000x4000 pixels, 548 convex polygons
Explicit Corridor Map: experiments
Performance• Setup
– NVIDIA GeForce 8800 GTX graphics card– Intel Core2 Quad CPU 2.4 GHz, 1 CPU used
• Experiments– City: 500x500 meter, 4000x4000 pixels, 548 convex polygons
time: 0.3s
Explicit Corridor Map: experiments
Supports large environments• E.g. 1 km2
• Millimeter precision– However, there must be at least
two pixels in between two obstacles to discover an edge
Explicit Corridor Map: recent work
Extension to 2.5D (multi-layered) environments• Technique
• Result (46 ms)
Multi-layered environment Partial medial axes for Li and LjConnection scene Updated medial axes for Li and Lj
Explicit Corridor Map: recent work
Handling dynamic changes• Technique for adding a point/line
• Result (1 – 2.7 ms per update)
A
Finding closest site Continue in 1 dir. w1 has been reached Updated VD (point) Updated VD (line)
Compare with the old approach
Disadvantages Implicit Corridor Map• More than linear storage (due to discrete sample points)• Non-exact representation• Additional parameters required
for local sampling density
Explicit Corridor Map: some thoughts
Examples of different topological spaces (plane, sphere, cylinder, torus, Möbius strip, Klein bottle)
The Corridor Map Method
Construction phase (offline)
• Build Explicit Corridor Map• Build kd-tree that stores
the ECM
Query phase (on-line)
• Construct indicative route– CMM: Medial axis
The Corridor Map Method
Query phase (on-line)
• Construct indicative route– CMM: Medial axis
• Compute a corridor• Compute a path
Construction phase (offline)
• Build Explicit Corridor Map• Build kd-tree that stores
the ECM
Distinguish three scales1. Macro (corridor)2. Meso (indicative route)3. Micro (local behavior)
Distinguish three scales1. Macro (corridor)2. Meso (indicative route)3. Micro (local behavior)
The Corridor Map Method
Query phase (on-line)1. Retract the start and goal to the medial axis
– Query the kd-tree
2. Connect the start and goal to the Corridor Map
3. Compute the shortest backbone path (using A*)
Explicit Corridor Map Corridor with its backbone pathQuery
The Corridor Map Method
Query phase (on-line)4. Compute the path
– While the corridor determines the character’s global path, forces determine its local path
– The force F(x)=Fa(x)+Fo(x) causes the character to accelerate, pulling it toward the goal. The variable x is the character’s position
– Using Newton's Second Law, we have F = Ma, where M = mass = 1 and a = acceleration
– Hence, the force F can be expressed as:
– Combining these expressions gives us:
A smooth path
smdtxdx /
2
2
2
)F(
)(F)(F/2
2
2
xx oasmdtxd
The Corridor Map Method
Query phase (on-line)4. Compute the path: forces
– The attraction force steers the character toward the goal
Fa(x) = , where f controls the magnitude
a(x) = attraction point: the furthest point on the backbone path whose disk encloses the character.
α(x)x
goal
Note on old approachThis is an discrete corridor instead of continuous explicit corridor.
Note on old approachThis is an discrete corridor instead of continuous explicit corridor.
The Corridor Map Method
Query phase (on-line)4. Compute the path: forces
– The boundary force keeps the character inside the corridorThis force is hidden inside the attraction force: (r=character’s radius)
f=0, when the character is positioned at its attraction point (i.e. d=0)f=∞, when the character touches the disk’s boundary
α(x)x
goal
R[t]
d
Note on old approachThis is an discrete corridor instead of continuous explicit corridor.
Note on old approachThis is an discrete corridor instead of continuous explicit corridor.
The Corridor Map Method
Query phase (on-line)4. Compute the path
– Solving the equation gives us the character’s positions– Cannot be done analytically
• revert to a numerical approximation
A smooth path
The Corridor Map Method
Choice of forces• Combining these forces and using disks was a bad choice• This is solved by the IRM, which uses Explicit Corridors
Comparison of their vector fields
Vector field: CMM force Vector field: IRM force
The Corridor Map Method: Examples
Query phase (on-line)4. Compute the path: forces
– Adding/changing forces leads to other “behavior”
Smooth path Short path Obstacle avoidance
Coherent groups Path variation Camera path
Obstacle avoidance (Helbing model) + path variation = crowd?
The Corridor Map Method: Examples
Query phase (on-line)4. Compute the path: forces
– Adding/changing forces leads to other “behavior”
Stealth-based path planning
Exploiting Explicit Corridors
The Corridor’s boundaries are given explicitly• Construction
Convenient representation• Small storage: linear in the number of samples (i.e. events)• Computation of closest points in O(1) time (on average)• Allows computing shortest minimum-clearance paths
Explicit Corridors: Obtaining clearance
Minimum clearance in explicit corridors• For each closest point cp, move cp toward its center point c• The displacement equals the desired clearance clmin
• Insert event point(s) if clmin > distance(c, cp)
c cp
Explicit Corridor Shrunk corridors Shrinking a corridor
Explicit Corridors: Shortest paths
Computing the shortest path• Construct a triangulation
– 2ith triangle: (li , ri , li+1) ; 2i+1th triangle: (ri , li+1 , ri+1)
– If the start [goal] is not included, add triangle (s, l1 , r1 ) [(ln , rn ,g)]
• Compute the shortest path– Funnel algorithm [Guibas et al. 1987]
Explicit Corridor Triangulation Shortest path
li ri
li+1 ri+1
s
g
Explicit Corridors: Shortest paths
Sketch of the Funnel algorithm• Funnel
– Tail: computed shortest path from start to apex
– Fan: 2 outward convex chains plus one diagonal
– The fan keeps track of all possible shortest paths
• Algorithm– Add diagonals iteratively while
updating the funnel
• Algorithm is linear in the number of diagonals
– (or events)
apex
start
goal
fan diagonal
tail
Explicit Corridors: Shortest paths
Computing the shortest minimum clearance path• Shrink the corridor
– Construction time: linear in the number of event points
• Compute the shortest path– Adjust Funnel algorithm to deal with circular arcs– Construction time: linear in the number of event points
Shortest pathLeft/right closest points Triangulation
Improving the CMM: IRM
Compute a smooth path: Indicative Route Method• Compute the shortest minimum-clearance path• Define the attraction force
– Pulls the character toward the goal
• Define the boundary force– Keeps the character inside the corridor
• Time-integrate the forces– Yields a smooth (C1-continous) path
The Query Phase: Experiments
Performance• Setup
– Intel Core2 Quad CPU 2.4 GHz, 1 CPU
• Experiments– City: 500x500 meter, 1.000 random queries
• Results (average query time)
The Query Phase: Experiments
Performance• Setup
– Intel Core2 Quad CPU 2.4 GHz, 1 CPU
• Experiments– City: 500x500 meter, 1 query
• Results (query time)– 2.8 ms
ECM (0.3s) Explicit corridor Shrunk corridor Triangulation Shortest path Smooth path
Integration in Second Life
Implementation in Second Life
Virtual World
Interface
Path planning on server:http request
Bitmap
Camera path
Conclusion
Advantages• Fast and flexible planner creates visually convincing paths• Computation of smooth, short minimum clearance paths• The algorithms run in linear time and are fast• The algorithms are simple
Open problems• Automatic annotation of the navigation mesh• Handling 3D spaces• Handling extensions• Handling character behavior
– E.g. shopping and beach behavior– Interaction between different
entities (human, car, bicycle)